Perfect numbers

"'I'll show you one more thing about perfect numbers," he said..."You can express them as the sum of consecutive natural numbers." Yoko Ogawa, The Housekeeper and the Professor, translated by Stephen Snyder

The examples that immediately follow the statement in the novel show that the perfect numbers 6, 28, and 496 are triangular numbers, that is, can be expressed as a sum of the form \(1+2+3+\cdots+n\).

This got me thinking about perfect numbers. For example,

Theorem: The reciprocals of the divisors of any perfect number sum to 2.
Proof: If the divisors of the perfect number \(N\) are \(1, d_2, d_3, \ldots, d_k\), and \(N\), then the sum of the reciprocals would be
\(1/1 + 1/d_2+ 1/d_3 \cdots + 1/d_k+ 1/N\). If we call this finite sum \(s\), then
\[Ns=N/1 + N/d_2+ N/d_3 \cdots + N/d_k+ N/N\]
\[ = N + d_k +\cdots +d_3+ d_2 + 1\]
\[= N +(\text{proper divisors of N})  = N+N = 2N\] Thus \( s=2\)

Theorem: Every even perfect number has the form \(N=(2^k -1)\cdot 2^{k-1}\), where \(2^k - 1\) is a Mersenne prime.
Proof:  Let \(N\) be an even perfect number. Then \(N\) can be written in the form \(N=2^{k-1} m\), where \(k>1 \)and \(m\) is odd.
Define \(\sigma(n)\) to be the sum of all positive divisors of \(n\).  In particular, \(\sigma(n)=2n\)  whenever \(n\) is perfect.
When \(a\) and \(b\) are relatively prime, \(\sigma(ab)=\sigma(a)\sigma(b)\)  because every divisor of \(ab\) can be uniquely written as the product of a divisor of \(a\) times a divisor of \(b\), so summing the divisors of \(ab\) can be accomplished by first computing \(\sigma(a)\)  and multiplying it by each of the divisors of \(b\), and summing those products.

Now \(\sigma(N)=2N=2^k m\) because \(N\) is perfect. By adding a finite geometric series with common ratio 2, we see that \(\sigma(2^{k-1})=2^k-1\), and we have

\(2^k m=  \sigma(2^{k-1}m) = \sigma(2^{k-1})\sigma(m)=(2^k-1)\sigma(m)\)  

Solving for \(\sigma(m)\), we get \(\sigma(m)=m+ \frac{m}{2^k-1} \)

From the definition of \(\sigma(m)\), \(m+ \frac{m}{2^k-1} \)  must represent the sum of all divisors of  \(m\), and in particular the fraction must be an integer.  But then \(\frac{m}{2^k-1}\) must itself divide \(m\), and as \(m\) clearly divides itself, \(m\) and \(\frac{m}{2^k-1} \)  must be all the divisors of \(m\).  Thus \(m=2^k-1\) must be prime.

Conversely, Theorem: Each Mersenne prime \(2^k -1\) gives the perfect number \(N=(2^k -1)\cdot 2^{k-1}\).
Proof: If \(2^k - 1\) is prime, then the divisors of \(N=(2^k -1)\cdot 2^{k-1}\) are \(1, 2, 2^2, \ldots, 2^{k-1},\) and also the product of any of those powers with the prime \(2^k - 1\). Summing all the proper divisors is the sum of two geometric series, each with common ratio 2. We get
\[\left(1+2+…+2^{k-1}\right) + \left(2^{k}-1\right) \left(1+2+…+2^{k-2}\right)\]
\[ = \left(2^k -1\right) + \left(2^{k}-1\right) \left(2^{k-1}-1\right)\]
\[= \left(2^k -1\right) \left(1+ 2^{k-1}-1\right) = N\]
We can see that every even perfect number is a triangular number, because \(N=(2^k -1)\cdot2^{k-1}\) has the form \(\frac{(n-1)n}{2}\), where \( n = 2^k\).

Euler evidently knew everything about even perfect numbers, but as far as I know, neither he nor anyone else has proven whether or not any odd perfect number exists.

I don't know whether an odd perfect number would need to be a triangular number. But the Professor only asserted that perfect numbers can be expressed as a sum of consecutive natural numbers. And of course any odd number \(O = 2n+1\) --including any odd perfect number-- can be expressed as the sum of two consecutive natural numbers: \(n + (n+1)\).

Inscribed Triangles and the Law of Sines

The Law of Sines follows from the following fact:  

Theorem: Any side of a triangle divided by the sine of the opposite angle gives the diameter of the circumscribing circle.

In other words, \( \frac{a}{\sin \alpha}, \frac{b}{\sin \beta}\), and \(\frac{c}{\sin \gamma} \) are all equal because each represents the same diameter.

The Law of Sines can be proven by using the fact that the area of a triangle is half the product of any two sides and the sine of the included angle. But such a proof gives no hint of the geometric interpretation of \( \frac{a}{\sin \alpha}\).

Here's a geometric argument for the theorem.

Case 1: First, we notice that if we have a right triangle with hypotenuse \(c\), then \(\sin\alpha =\frac{a}{c}\), \(\sin\beta =\frac{b}{c}\), and \(\sin\gamma =\sin 90^{\circ}=1\), so all three ratios  \( \frac{a}{\sin \alpha}, \frac{b}{\sin \beta}\), and \(\frac{c}{\sin \gamma} \)  are equal to \(c\), the hypotenuse. And because an inscribed right angle subtends an arc of \(180^{\circ}\), the hypotenuse coincides with a diameter. Thus the theorem is true for all angles in any right triangle.

Case 2: Now suppose that \(\Delta ABC\) is a triangle with \(\alpha =\angle CAB\) an acute angle. Draw the diameter through \(B\) to the point \(D\), and draw the segment from \(D\) to \(C\).

\(\Delta DBC\) is a right triangle inscribed in the same circle and shares the side of length \(a\) with \(\Delta ABC\). The angle at \(D\) subtends the same arc as the angle at \(A\), so the angles are congruent. By Case 1, \(\frac{a}{\sin\alpha}\) is the diameter of the circumscribing circle. Thus the theorem holds for any acute angle in any triangle.

Case 3: Now suppose that  \(\Delta ABC\) is a triangle with \(\alpha =\angle CAB\) an obtuse angle. Choose \(D\) so it lies on the arc of the circle subtended by \(\angle CAB\). Then \(\Delta DBC\) is inscribed in the same circle as \(\Delta ABC\) and shares the side of length \(a\).

Because \(\angle CAB\) and \(\angle CDB\) subtend arcs that sum to \(360^{\circ}\), they are supplementary angles. In particular, \(\angle CDB\) is acute, so the diameter of the circumscribing circle is \(\frac{a}{\sin(180^{\circ}-\alpha)}\). And because \(\sin(180^{\circ}-\alpha) = \sin\alpha\), we conclude that the diameter of the circumscribing circle is  \( \frac{a}{\sin \alpha}\).

Thus, whether \(\alpha\) is a right angle, acute angle, or obtuse angle, the ratio  \( \frac{a}{\sin \alpha}\) gives the diameter of the circumscribing circle.

Higher Education Alignment with the Common Core

The August 29, 2014 letter from California's higher education top administrators  announced that "the a-g requirements for CSU and UC admission, specifically areas ‘b’ (English) and ‘c’ (Mathematics), have been updated to align with the Common Core standards."

How that alignment will look is not specified in the letter.

As of today (9/9/14), the UC Mathematics ("c") subject requirements listed publicly do not show alignment with the Common Core State Standards. Instead, they still show expectations of California standards that existed before the CCSSM. For example, in item 2 of Course requirements, "The content for these courses will usually be drawn from the Common Core State Standards for Mathematics [PDF]. While these standards can be a useful guide, coverage of all items in the standards is not necessary for the specific purpose of meeting the 'c' subject requirement....The ICAS Statement of Competencies in Mathematics can provide guidance in selecting topics that require in-depth study." [Emphasis mine.]

A concern for California community colleges is that the alignment to the CCSSM might become what was proposed by the UC Board of Admissions and Relations with Schools (BOARS) in 2013. In July, BOARS wrote that “… the basic mathematics of the CCSSM can appropriately be used to define the minimal level of mathematical competence that all incoming UC students should demonstrate...As such, BOARS expects that the Transferable Course Agreement Guidelines will be rewritten to clarify that the prerequisite mathematics for transferable courses should align with the college-ready content standards of the CCSSM.”

BOARS clarified (December 2013) that “… going forward, all students must complete the basic mathematics defined by the college-ready standards of the Common Core State Standards for Mathematics (CCSSM) prior to enrolling in a UC-transferable college mathematics or statistics course.”

The college-ready standards of the CCSSM are simply all the non-plus standards. As written in the CCSSM, “The higher mathematics standards specify the mathematics that all students should study in order to be college and career ready. Additional mathematics that students should learn in preparation for advanced courses, such as calculus, advanced statistics, or discrete mathematics, is indicated by a plus symbol (+). All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students.” [Emphasis mine]

Thus BOARS has twice stated that it expects all UC students to have all the CCSSM non-plus standards as prerequisite to any course that could receive UC credit.

But what undermines BOARS's credibility is its assessment of how the ICAS statement of competencies and the CCSSM content standards compare. In the opening paragraph of the BOARS July letter: "The most recent version of the ICAS mathematical competency statement makes clear the close alignment between it and the CCSSM. Both define the mathematics that all students should study in order to be college ready." [Emphasis mine.]

In actuality, what ICAS considers essential math content for all students is only a small subset of what the CCSSM specify as necessary. The ICAS document lists four sets of possible high school math topics. The first is Part 1: Essential areas of focus for all entering college students. Appendix B of the ICAS document explicitly shows how the CCSSM include not only the math topics of Part 1 but also the math topics of Parts 2, 3, and 4, which are areas of focus for students in quantitative majors or are areas of focus considered desirable but not essential.

Cosine of 72 degrees (and constructing a regular pentagon)

By using (say) DeMoivre's theorem, we have that \( \left( \cos\frac{2 \pi}{5}+ i \sin\frac{2\pi}{5} \right)^5=1\)
Expanding the left side as the fifth power of a binomial, equating the imaginary parts on both sides of the equation, and then replacing  \( \sin^2 72^\circ\) with  \(1- \cos^2 72^\circ \)

 \( i\sin 72^\circ \left(5 \cos^4 72^\circ+10 \cos^2 72^\circ i^2 \sin ^2 72 ^\circ+ i^4 \sin^4 72^\circ \right)=0i\)

 \(16 \cos^4 72^\circ - 12 \cos^2 72^\circ  + 1=0\)

Solving this quadratic in \( \cos^2 72^\circ \), we get

 \[ \cos^2 72^\circ  = \frac{12 \pm  \sqrt{80} } {32}   \]

so

 \[ \cos^2 72^\circ  = \frac{6 \pm 2 \sqrt{5} } {16}=\frac{\left( \sqrt{5}\pm 1 \right)^2}{4^2}   \]

 \[ \cos 72^\circ  = \pm \frac{\sqrt{5} \pm 1}{4}  \]

where we can choose the correct value of the four possible values by noting that, because 72° is between 45° and 90°,  \( \cos 72^\circ \) must lie between  \(1 / \sqrt{2}\) and 0. Because  \( \cos 72^\circ \) is positive, we choose the "+" before the fraction, and because  \( \cos 72^\circ \) is less than \( 1 / \sqrt{2}\), which in turn is less than \(\frac{\sqrt{5}+1}{4}\), we choose the "-" in the numerator:

\[ \cos 72^\circ = \frac{\sqrt{5}-1}{4} \]

Constructing a regular pentagon

So we can construct \( \cos 72^\circ\). For example, the diagonal of a 1-by-2 rectangle is \(\sqrt{5}\). We could cut off one unit from a segment of length \(\sqrt{5}\), then divide the segment of length \(\sqrt{5}-1\) into four pieces of length \( \frac{\sqrt{5} -1} {4} \). (Or we could construct the appropriate solution to the equation \( 4x^2 +2x -1 = 0 \). See my post on Solving quadratic equations via geometric construction.)

Construct a unit circle centered at O, and construct a radius \(\overline{OA}\).  Construct the point B on \(\overline{OA}\) so that \(\overline{OB}\) has length  \( \cos 72^\circ\). If C is a point on the circle so that \(\overline{BC}\) is perpendicular to \(\overline{OA}\), then \(\angle COA\) is a 72° angle, and both A and C are vertices of a regular pentagon inscribed in the circle.

What Math is Needed by All?

The (California version of the) Common Core State Standards in mathematics purport to be what all students need to be college and career ready.

The quantifier "all" in this context indicates that the math content should be the intersection (over all students) of math a student needs to be ready to begin college (or begin a career). Critics of the CCSSM who decry that the standards are not enough to prepare a student for an elite university such as Stanford are missing the point. The intent of the CCSS was never to include the union (over all students) of the math that a student needs to succeed in college. (And if the CCSS could provide all the math and English Language Arts that Stanford students need, then Stanford would not deserve its status as an elite school.)

And what do all students need? In 2013, the National Center on Education and the Economy released a study What Does It Really Mean to Be College and Work Ready?, reporting on both mathematics and English literacy. The report says, "Mastery of Algebra II is widely thought to be a prerequisite for success in college and careers. Our research shows that that is not so... Based on our data, one cannot make the case that high school graduates must be proficient in Algebra II to be ready for college and careers."

California's Intersegmental Committee of the Academic Senates (ICAS) represents the faculty academic senates of the three CA systems of higher education: the University of California (UC), the California State University (CSU), and the California Community College (CCC) system. The ICAS Statement on Competencies in Mathematics Expected of Entering College Students, revised in 2013, describes a number of mathematical topics that are or could be taught in high schools.

The ICAS competency statement describes mathematical subject matter in four categories: Part 1: Essential areas of focus for all entering college students, Part 2: Desirable areas of focus for all entering college students, Part 3: Essential areas of focus for students in quantitative majors, and Part 4: Desirable areas of focus for students in quantitative majors.

The mathematics that the CCSSM describe as what all students need should presumably match with what the ICAS statement describes as "essential" and lists in Part 1. But although the UC Board of Admissions and Relations with Schools (BOARS) states there is "close alignment" between the CCSS and the ICAS statement, the ICAS statement makes clear that there are many CCSS that are not "essential" but rather merely desirable or for only some students. Appendix B of the ICAS statement explicitly shows where Part 2, 3, and 4 areas of math are found in the CCSS (and NCTM standards).

And the Interim Environmental Scan Report to The Common Assessment Initiative Steering Committee has in  Appendix B a Table that shows a number of CCSS that do not occur at all in the ICAS statement.

Here are examples of CCSSM topics that might surprise some community college math faculty, especially those who believe that intermediate algebra as currently taught will be sufficient to cover all the CCSSM.

• Probability:  sample spaces, independent events, conditional probability, permutations and combinations; analyzing decisions and strategies using probability
• Statistics: assessing the fit of a function by plotting and analyzing residuals; interpreting the correlation coefficient of a linear model in context; normal distributions, random samples, estimating population parameters, simulations, using probability to make decisions
• Transformational geometry: congruence defined in terms of rigid motion; similarity defined in terms of dilations and rigid motions
• Trigonometry: trig ratios, special angles, 6 trig functions of real numbers; modeling periodic phenomena, proof and use of the Pythagorean trig identity \( \cos^2 \theta + \sin^2 \theta = 1 \)

Schizophrenic Common Core Supporter

Back in 2012 Sol Garfunkel wrote "I feel like a schizophrenic. I truly think that the Common Core State Standards for Mathematics (CCSSM) are a disaster...So why do I feel like a schizophrenic? Because I am at the same time working to make the implementation of the CCSSM be as effective as possible!"

As mathematician Keith Devlin has emphasized, the heart of the CCSSM is the set of 8  standards of Mathematical Practice:

• MP1. Make sense of problems and persevere in solving them.
• MP2. Reason abstractly and quantitatively.
• MP3. Construct viable arguments and critique the reasoning of others.
• MP4. Model with mathematics.
• MP5. Use appropriate tools strategically.
• MP6. Attend to precision.
• MP7. Look for and make use of structure.
• MP8. Look for and express regularity in repeated reasoning.

It would be hard to imagine that any mathematician or math educator would not applaud these standards. And these standards, the key to the CCSSM and presented at the start of each set of grade level standards, are rarely if ever mentioned in the attacks on the CCSSM.

Much of the resistance to the CCSS is political: the Democratic President of the United States has endorsed the CCSS, so there is automatic opposition from the Tea Party, Republicans, and Libertarians, who argue that the CCSS is a federal program. But although President Obama is giving incentives for states to adopt the CCSS, the standards are the result of 48 state governors and secretaries of education agreeing to cooperate to create educational standards that would be consistent across state lines.

The resistance from the classroom teachers is understandable because they will be held accountable to how their students will do on the CCSS standardized testing. But the standardized testing that will be used is not part of the CCSS but rather is being created by SBAC or PARCC, consortia created to write CCSS assessments. That is, although the news media report teacher opposition to the CCSS, the teachers' actual objection is to the assessments and how they will be used.

The widely seen mocking and vilification of CCSS lessons by the public also confuse the CCSS with methods for testing students for mathematical proficiency. The CCSS explicitly require that students master the standard algorithms that critics mistakenly say are "real math" and missing from the CCSS. But significantly, the CCSS also require (MP1) that students can make sense of the mathematical tasks they are performing.

I think the CCSSM grossly overshoot the mark when trying to specify the math that all students need to be college and career ready. But like Sol Garfunkel, I think we should simultaneously embrace the CCSS and work to improve them.

Common Core Goes to College

The New America Foundation’s position paper by Lindsey Tepe gives recommendations for how higher education can support the Common Core State Standards. However, this paper and related articles in the Chronicle  and Hechinger Report  miss the most important way for higher education to support the CCSS, namely, to work to repair or ameliorate the existing flaws in the CCSS. 

An implicit assumption in Tepe's paper is that the CCSS have successfully captured what all students need to be college and career ready. If the assumption is false, the paper is advocating moves to change higher education to accommodate inappropriate standards, changes that could harm students and impede their paths to college degrees.

The CCSS have missed the mark at what is necessary for all students to succeed in college.

Many of the non-plus CCSS are currently introduced to students in credit-bearing courses of baccalaureate granting institutions. That is, the CCSS overshoots what is needed to be ready for college and includes topics that are part of what some college students need to learn while in college.

The intent of the CCSS was to help get students college (and career) ready. It is an abuse of the CCSS to use those standards as an opportunity for colleges and universities to raise admissions and/or degree requirements, and that abuse will work against the goal of giving more students the opportunity to earn college degrees.

Math Initiatives for Student Success

The LearningWorks paper Changing Equations: How Community Colleges Are Re-thinking College Readiness in Math, written by Pamela Burdman, is a nice summary of current initiatives attempting to help capable students negotiate developmental math needs to succeed in transfer-level mathematics.

Much of the paper discusses the strategy of alternative pathways. In this strategy, students pass a course that is identical to, or has the same content and rigor of, accepted transfer math courses, but instead of first passing an intermediate algebra course, the students take a math course designed specifically to prepare them for the transfer course—that preparatory course omits some standard topics of intermediate algebra which are not necessary to succeed in the transfer math course.

The initial data on alternative pathways, some cited in Changing Equations, show that a much higher percentage of students initially placed in a developmental math course can pass a transfer level math course following an alternative pathway than by following the traditional chain of prerequisites. 

But both the University of California and the California State University systems require that intermediate algebra be a prerequisite for any transferable course. Keeping the intermediate algebra prerequisite based on the data that have shown success in intermediate algebra is a predictor of college success is, as pointed out in Changing Equations, following the error of confusing correlation with causation, and in fact the widespread practice of requiring success in intermediate algebra (a.k.a. Algebra 2) as a admissions requirement virtually guarantees the high correlation that has been often noted.

JMM 2014

Over 6400 mathematicians descended upon Baltimore January 15-18 for the 2014 Joint Mathematics Meetings. Sessions included current research in math, discussions on pedagogy, content, collaborations across institutions, social events, and more.

The first session on Wednesday 15 January was the MAA Minority Chairs committee meeting at 7:00 am, although there were actually some short courses, workshops, AMS council meetings and MAA Board of Governors meeting on the preceding Monday and Tuesday. And there were dozens of contributed paper sessions throughout the morning and the rest of the day.

The JMM unveiled the theme of the 2014 Math Awareness Month (April 2014): Mathematics, Magic, & Mystery (http://www.mathaware.org/). On each day of April 2014 a new square of the Activity Calendar goes live, giving access to mathematical puzzles and magic. (Once opened, the resources are to be kept available for as long as the AMS exists.)

JMM2014 also included a panel session launching TPSE Math: Transforming Post-Secondary Education in Mathematics (@tpsem, http://www.tpsemath.org/). The project is sponsored by the Carnegie Foundation of New York and the Alfred P. Sloan Foundation.

One of the sessions on the last day was The Public Face of Mathematics http://blogs.ams.org/jmm2014/2014/01/18/the-public-face-of-mathematics/.  The panel was organized by mathemagician Art Benjamin (http://www.math.hmc.edu/~benjamin/) and included "Math Guy" Keith Devlin (http://www.stanford.edu/~kdevlin/), NY Times columnist Steven Strogatz (http://www.stevenstrogatz.com/), mathbabe Cathy O'Neill (http://mathbabe.org/), freelance journalist Tom Siegfried (http://www.sciencenoise.org/), and US Congressman Jerry McNerny (http://mcnerney.house.gov/).